Puree all dressing ingredients in a blender or small food processor until smooth. Season to taste with salt and set aside. Grill romaine, radicchio and asparagus on a well-oiled grill over medium heat for 5 to 10 minutes, turning frequently, until lightly charred. (Radicchio will cook quickly, asparagus will take longer.) Remove from grill and let cool slightly. Cut radicchio and romaine into 1/2-inch strips, then quarter asparagus spears lengthwise and cut into bite-size strips; place all in a medium bowl. Drizzle with dressing and toss lightly to coat. Sprinkle with Parmesan and toss again very lightly.

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